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Discrete & Continuous Dynamical Systems
July 2012 , Volume 32 , Issue 7
Special issue on Dynamical Systems II,
Denton 2009; Conference Proceedings
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2012, 32(7): i-iii
doi: 10.3934/dcds.2012.32.7i
+[Abstract](2346)
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Abstract:
The conference "Dynamical Systems II, Denton 2009" held in the University of North Texas in Denton from May 17, 2009 through May 23, 2009 gathered approximately forty participants working on various subbranches of dynamical systems such as holomorphic and conformal dynamics, transcendental dynamics, random dynamical systems, thermodynamic formalism, and iterated function systems, including random behavior of deterministic systems, the theory of fractal sets, including dimension theory. Apart from stimulating, highly informative mathematical talks delivered by nearly all participants of the conference, its outcome resulted also in thirteen outstanding research articles presented in this volume. The subject of these papers varies from author to author reflecting their scientific interests. The articles were written by A. Badeńska (Warsaw University of Technology), D. Hensley (Texas A&M), P. Haissinsky (Université de Provence) , M. Kesseboehmer (University of Bremen), D. Mayer (TU Clausthal), E. Mihailescu (Romanian Academy), T. Mühlenbruch (FernUniversität in Hagen), F. Naud (Université d'Avignon), S. Munday (University of St Andrews), K. Pilgrim (Indiana University), Mario Roy (York University), H. H. Rugh, D. Simmons (University of North Texas), B. Stratmann (University of Bremen), F. Strömberg (TU Darmstadt), H. Sumi (University of Osaka), and M. Urbański (University of North Texas).
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The conference "Dynamical Systems II, Denton 2009" held in the University of North Texas in Denton from May 17, 2009 through May 23, 2009 gathered approximately forty participants working on various subbranches of dynamical systems such as holomorphic and conformal dynamics, transcendental dynamics, random dynamical systems, thermodynamic formalism, and iterated function systems, including random behavior of deterministic systems, the theory of fractal sets, including dimension theory. Apart from stimulating, highly informative mathematical talks delivered by nearly all participants of the conference, its outcome resulted also in thirteen outstanding research articles presented in this volume. The subject of these papers varies from author to author reflecting their scientific interests. The articles were written by A. Badeńska (Warsaw University of Technology), D. Hensley (Texas A&M), P. Haissinsky (Université de Provence) , M. Kesseboehmer (University of Bremen), D. Mayer (TU Clausthal), E. Mihailescu (Romanian Academy), T. Mühlenbruch (FernUniversität in Hagen), F. Naud (Université d'Avignon), S. Munday (University of St Andrews), K. Pilgrim (Indiana University), Mario Roy (York University), H. H. Rugh, D. Simmons (University of North Texas), B. Stratmann (University of Bremen), F. Strömberg (TU Darmstadt), H. Sumi (University of Osaka), and M. Urbański (University of North Texas).
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2012, 32(7): 2375-2402
doi: 10.3934/dcds.2012.32.2375
+[Abstract](2273)
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Abstract:
We consider hyperbolic meromorphic functions of the following form $f(z)=R\circ\exp(z)$, where $R$ is a non-constant rational function, satisfying so-called rapid derivative growth condition. We study several types of conjugacies in this class and prove a~measure rigidity theorem in the case when $f$ has a logarithmic tract over $\infty$ and under some additional assumptions.
We consider hyperbolic meromorphic functions of the following form $f(z)=R\circ\exp(z)$, where $R$ is a non-constant rational function, satisfying so-called rapid derivative growth condition. We study several types of conjugacies in this class and prove a~measure rigidity theorem in the case when $f$ has a logarithmic tract over $\infty$ and under some additional assumptions.
2012, 32(7): 2403-2416
doi: 10.3934/dcds.2012.32.2403
+[Abstract](2668)
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Abstract:
In previous work, a class of noninvertible topological dynamical systems $f: X \to X$ was introduced and studied; we called these topologically coarse expanding conformal systems. To such a system is naturally associated a preferred quasisymmetry (indeed, snowflake) class of metrics in which arbitrary iterates distort roundness and ratios of diameters by controlled amounts; we called this metrically coarse expanding conformal. In this note we extend the class of examples to several more familiar settings, give applications of our general methods, and discuss implications for the computation of conformal dimension.
In previous work, a class of noninvertible topological dynamical systems $f: X \to X$ was introduced and studied; we called these topologically coarse expanding conformal systems. To such a system is naturally associated a preferred quasisymmetry (indeed, snowflake) class of metrics in which arbitrary iterates distort roundness and ratios of diameters by controlled amounts; we called this metrically coarse expanding conformal. In this note we extend the class of examples to several more familiar settings, give applications of our general methods, and discuss implications for the computation of conformal dimension.
2012, 32(7): 2417-2436
doi: 10.3934/dcds.2012.32.2417
+[Abstract](3205)
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Abstract:
We survey the dynamical systems side of the theory of continued fractions and touch on some of the frontiers of the subject. Ergodic theory plays a role. The work of Baladi and Vallée is discussed. Power series methods that allow for the computation of various numbers such as the Hausdorff dimension of a continued fraction Cantor set, or the Wirsing constant of a particular continued fraction algorithm, to high accuracy, are also discussed.
We survey the dynamical systems side of the theory of continued fractions and touch on some of the frontiers of the subject. Ergodic theory plays a role. The work of Baladi and Vallée is discussed. Power series methods that allow for the computation of various numbers such as the Hausdorff dimension of a continued fraction Cantor set, or the Wirsing constant of a particular continued fraction algorithm, to high accuracy, are also discussed.
2012, 32(7): 2437-2451
doi: 10.3934/dcds.2012.32.2437
+[Abstract](3125)
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Abstract:
In this paper we give a detailed measure theoretical analysis of what we call sum-level sets for regular continued fraction expansions. The first main result is to settle a recent conjecture of Fiala and Kleban, which asserts that the Lebesgue measure of these level sets decays to zero, for the level tending to infinity. The second and third main results then give precise asymptotic estimates for this decay. The proofs of these results are based on recent progress in infinite ergodic theory, and in particular, they give non-trivial applications of this theory to number theory. The paper closes with a discussion of the thermodynamical significance of the obtained results, and with some applications of these to metrical Diophantine analysis.
In this paper we give a detailed measure theoretical analysis of what we call sum-level sets for regular continued fraction expansions. The first main result is to settle a recent conjecture of Fiala and Kleban, which asserts that the Lebesgue measure of these level sets decays to zero, for the level tending to infinity. The second and third main results then give precise asymptotic estimates for this decay. The proofs of these results are based on recent progress in infinite ergodic theory, and in particular, they give non-trivial applications of this theory to number theory. The paper closes with a discussion of the thermodynamical significance of the obtained results, and with some applications of these to metrical Diophantine analysis.
2012, 32(7): 2453-2484
doi: 10.3934/dcds.2012.32.2453
+[Abstract](2826)
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Abstract:
In this paper we extend the transfer operator approach to Selberg's zeta function for cofinite Fuchsian groups to the Hecke triangle groups $G_q,\, q=3,4,\ldots$, which are non-arithmetic for $q\not= 3,4,6$. For this we make use of a Poincar\'e map for the geodesic flow on the corresponding Hecke surfaces, which has been constructed in [13], and which is closely related to the natural extension of the generating map for the so-called Hurwitz-Nakada continued fractions. We also derive functional equations for the eigenfunctions of the transfer operator which for eigenvalues $\rho =1$ are expected to be closely related to the period functions of Lewis and Zagier for these Hecke triangle groups.
In this paper we extend the transfer operator approach to Selberg's zeta function for cofinite Fuchsian groups to the Hecke triangle groups $G_q,\, q=3,4,\ldots$, which are non-arithmetic for $q\not= 3,4,6$. For this we make use of a Poincar\'e map for the geodesic flow on the corresponding Hecke surfaces, which has been constructed in [13], and which is closely related to the natural extension of the generating map for the so-called Hurwitz-Nakada continued fractions. We also derive functional equations for the eigenfunctions of the transfer operator which for eigenvalues $\rho =1$ are expected to be closely related to the period functions of Lewis and Zagier for these Hecke triangle groups.
2012, 32(7): 2485-2502
doi: 10.3934/dcds.2012.32.2485
+[Abstract](2435)
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Abstract:
The dynamics of endomorphisms (i.e non-invertible smooth maps) presents many significant differences from that of diffeomorphisms, as well as from the dynamics of expanding maps. There are numerous concrete examples of hyperbolic endomorphisms. Many methods cannot be used here due to overlappings in the fractal set and to the existence of (possibly infinitely) many local unstable manifolds going through the same point. First we will present the general problems and explain how to construct certain useful limit measures for atomic measures supported on various prehistories. These limit measures are in many cases shown to be equal to certain equilibrium measures for Hölder potentials. We obtain thus an analogue of the SRB measure, namely an inverse SRB measure in the case of a hyperbolic repeller, or of an Anosov endomorphism. We study then the 1-sided Bernoullicity (or lack of it) for certain measures invariant to endomorphisms, and give a Classification Theorem for the ergodic and metric types of behaviour of perturbations of a class of maps on their respective basic sets, in terms of the values of the stable dimension. We give also relations between thermodynamic formalism and fractal dimensions (Hausdorff dimension of stable/unstable intersections with basic sets, stable/unstable box dimensions, dimension of the global unstable set for endomorphisms). Applications to certain nonlinear evolution models are also given in the end.
The dynamics of endomorphisms (i.e non-invertible smooth maps) presents many significant differences from that of diffeomorphisms, as well as from the dynamics of expanding maps. There are numerous concrete examples of hyperbolic endomorphisms. Many methods cannot be used here due to overlappings in the fractal set and to the existence of (possibly infinitely) many local unstable manifolds going through the same point. First we will present the general problems and explain how to construct certain useful limit measures for atomic measures supported on various prehistories. These limit measures are in many cases shown to be equal to certain equilibrium measures for Hölder potentials. We obtain thus an analogue of the SRB measure, namely an inverse SRB measure in the case of a hyperbolic repeller, or of an Anosov endomorphism. We study then the 1-sided Bernoullicity (or lack of it) for certain measures invariant to endomorphisms, and give a Classification Theorem for the ergodic and metric types of behaviour of perturbations of a class of maps on their respective basic sets, in terms of the values of the stable dimension. We give also relations between thermodynamic formalism and fractal dimensions (Hausdorff dimension of stable/unstable intersections with basic sets, stable/unstable box dimensions, dimension of the global unstable set for endomorphisms). Applications to certain nonlinear evolution models are also given in the end.
2012, 32(7): 2503-2520
doi: 10.3934/dcds.2012.32.2503
+[Abstract](2778)
+[PDF](245.1KB)
Abstract:
Certain subsets of limit sets of geometrically finite Fuchsian groups with parabolic elements are considered. It is known that Jarník limit sets determine a "weak multifractal spectrum" of the Patterson measure in this situation. This paper will describe a natural generalisation of these sets, called strict Jarník limit sets, and show how these give rise to another weak multifractal spectrum. Number-theoretical interpretations of these results in terms of continued fractions will also be given.
Certain subsets of limit sets of geometrically finite Fuchsian groups with parabolic elements are considered. It is known that Jarník limit sets determine a "weak multifractal spectrum" of the Patterson measure in this situation. This paper will describe a natural generalisation of these sets, called strict Jarník limit sets, and show how these give rise to another weak multifractal spectrum. Number-theoretical interpretations of these results in terms of continued fractions will also be given.
2012, 32(7): 2521-2531
doi: 10.3934/dcds.2012.32.2521
+[Abstract](2641)
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Abstract:
We define a natural space of transfer operators related to holomorphic contraction systems. We show that the classical upper bounds on the Ruelle eigenvalue sequence are optimal for a dense set of transfer operators. A similar statement is derived for Perron-Frobenius operators related to uniformly expanding piecewise real analytic interval maps. The proof is based on potential theory.
We define a natural space of transfer operators related to holomorphic contraction systems. We show that the classical upper bounds on the Ruelle eigenvalue sequence are optimal for a dense set of transfer operators. A similar statement is derived for Perron-Frobenius operators related to uniformly expanding piecewise real analytic interval maps. The proof is based on potential theory.
2012, 32(7): 2533-2551
doi: 10.3934/dcds.2012.32.2533
+[Abstract](2660)
+[PDF](422.6KB)
Abstract:
We introduce a new variation of Bowen's formula for conformal graph directed Markov systems (a.k.a. CGDMSs). This new variation applies to a very large collection of non-irreducible systems and is shown to coincide with the well-known Bowen's formula that holds for all finite or finitely irreducible CGDMSs (cf. [2], [4] and [1]). We further show that the original version of Bowen's formula may not hold even for non-irreducible CGDMSs whose components are IFSs, justifying thereby the introduction of a new variation. This answers two questions that were raised by Ghenciu and Mauldin in [1]. Their third question is also %partially tackled. addressed. Indeed, we prove that Ghenciu and Mauldin's conjecture about the finiteness parameters of the partition functions of the pressure is false even within the class of irreducible systems.
We introduce a new variation of Bowen's formula for conformal graph directed Markov systems (a.k.a. CGDMSs). This new variation applies to a very large collection of non-irreducible systems and is shown to coincide with the well-known Bowen's formula that holds for all finite or finitely irreducible CGDMSs (cf. [2], [4] and [1]). We further show that the original version of Bowen's formula may not hold even for non-irreducible CGDMSs whose components are IFSs, justifying thereby the introduction of a new variation. This answers two questions that were raised by Ghenciu and Mauldin in [1]. Their third question is also %partially tackled. addressed. Indeed, we prove that Ghenciu and Mauldin's conjecture about the finiteness parameters of the partition functions of the pressure is false even within the class of irreducible systems.
2012, 32(7): 2553-2564
doi: 10.3934/dcds.2012.32.2553
+[Abstract](2062)
+[PDF](205.1KB)
Abstract:
We consider random conformal repellers. We show how to apply Bowen's formula for the Hausdorff dimension in this context and prove smoothness of the dimension with respect to parameters. The present article is essentially an extract of [11]. Our aim here is to emphasize the ideas and mechanisms behind rather than mathematical rigor.
We consider random conformal repellers. We show how to apply Bowen's formula for the Hausdorff dimension in this context and prove smoothness of the dimension with respect to parameters. The present article is essentially an extract of [11]. Our aim here is to emphasize the ideas and mechanisms behind rather than mathematical rigor.
2012, 32(7): 2565-2582
doi: 10.3934/dcds.2012.32.2565
+[Abstract](3687)
+[PDF](444.9KB)
Abstract:
The purpose of this paper is to give a clean formulation and proof of Rohlin's Disintegration Theorem [7]. Another (possible) proof can be found in [6]. Note also that our statement of Rohlin's Disintegration Theorem (Theorem 2.1) is more general than the statement in either [7] or [6] in that $X$ is allowed to be any universally measurable space, and $Y$ is allowed to be any subspace of standard Borel space.
Sections 1 - 4 contain the statement and proof of Rohlin's Theorem. Sections 5 - 7 give a generalization of Rohlin's Theorem to the category of $\sigma$-finite measure spaces with absolutely continuous morphisms. Section 8 gives a less general but more powerful version of Rohlin's Theorem in the category of smooth measures on $C^1$ manifolds. Section 9 is an appendix which contains proofs of facts used throughout the paper.
The purpose of this paper is to give a clean formulation and proof of Rohlin's Disintegration Theorem [7]. Another (possible) proof can be found in [6]. Note also that our statement of Rohlin's Disintegration Theorem (Theorem 2.1) is more general than the statement in either [7] or [6] in that $X$ is allowed to be any universally measurable space, and $Y$ is allowed to be any subspace of standard Borel space.
Sections 1 - 4 contain the statement and proof of Rohlin's Theorem. Sections 5 - 7 give a generalization of Rohlin's Theorem to the category of $\sigma$-finite measure spaces with absolutely continuous morphisms. Section 8 gives a less general but more powerful version of Rohlin's Theorem in the category of smooth measures on $C^1$ manifolds. Section 9 is an appendix which contains proofs of facts used throughout the paper.
2012, 32(7): 2583-2589
doi: 10.3934/dcds.2012.32.2583
+[Abstract](3129)
+[PDF](302.7KB)
Abstract:
In [13] there is a survey of several methods of proof that the Julia set of a rational or entire function is the closure of the repelling cycles, along with a discussion of which of those methods can and cannot be extended to the case of semigroups. In particular that paper presents an elementary proof based on the ideas of [11] that the Julia set of either a non-elementary rational or entire semigroup is the closure of the set of repelling fixed points. This paper serves as a brief follow up to [13] by showing that the ideas of [3] can also be used to provide an elementary proof for the semigroup case. It also touches upon some key differences between the dynamics of iteration and the dynamics of semigroups.
In [13] there is a survey of several methods of proof that the Julia set of a rational or entire function is the closure of the repelling cycles, along with a discussion of which of those methods can and cannot be extended to the case of semigroups. In particular that paper presents an elementary proof based on the ideas of [11] that the Julia set of either a non-elementary rational or entire semigroup is the closure of the set of repelling fixed points. This paper serves as a brief follow up to [13] by showing that the ideas of [3] can also be used to provide an elementary proof for the semigroup case. It also touches upon some key differences between the dynamics of iteration and the dynamics of semigroups.
2012, 32(7): 2591-2606
doi: 10.3934/dcds.2012.32.2591
+[Abstract](3032)
+[PDF](327.5KB)
Abstract:
We estimate the Bowen parameters and the Hausdorff dimensions of the Julia sets of expanding finitely generated rational semigroups. We show that the Bowen parameter is larger than or equal to the ratio of the entropy of the skew product map $\tilde{f}$ and the Lyapunov exponent of $\tilde{f}$ with respect to the maximal entropy measure for $\tilde{f}$. Moreover, we show that the equality holds if and only if the generators are simultaneously conjugate to the form $a_{j}z^{\pm d}$ by a M\"{o}bius transformation. Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that the Bowen parameter is strictly larger than $2$.
We estimate the Bowen parameters and the Hausdorff dimensions of the Julia sets of expanding finitely generated rational semigroups. We show that the Bowen parameter is larger than or equal to the ratio of the entropy of the skew product map $\tilde{f}$ and the Lyapunov exponent of $\tilde{f}$ with respect to the maximal entropy measure for $\tilde{f}$. Moreover, we show that the equality holds if and only if the generators are simultaneously conjugate to the form $a_{j}z^{\pm d}$ by a M\"{o}bius transformation. Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that the Bowen parameter is strictly larger than $2$.
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